91Ó°ÊÓ

Closed-cell extruded polystyrene foam, or XPS, is a trademarked brand of Styrofoam. Blue Board is another name for this foam, which is used for building insulation, thermal insulation, and water barriers.

The polymer polystyrene, with styrene as the repeating monomer, is used to make Styrofoam. Only carbon and hydrogen make up styrene.

Use the expression for buoyant force to determine the required mass.

The buoyant force is expressed as follows:

$F_B={\mathrm{ÒÏ}}_f{V}_{f}g$

Here, ${\mathrm{ÒÏ}}_f$is density of the fluid and$v_f$ is volume of displaced fluid which is equal to volume of Styrofoam sphere.

For the mass that can hang without sinking, the downward gravitational force must be balanced by the upward buoyant force.

In this case, the net force on the mass is expressed as follows:

$F_{net}=F_B-m_{s}g$

Here, $F_B$is buoyant force $m_s$is mass of Styrofoam sphere, and$g$is acceleration due to gravity.

Combine the above two relations.

$mg=\left({\mathrm{ÒÏ}}_f{V}_{f}g\right)-m_{s}g$

Here, $m$is mass of the object.

Volume of Styrofoam sphere is,

$V_f=\frac{4}{3}{\mathrm{Ï€°ù}}^3$

Mass of the Styrofoam sphere is,

$m_s={\mathrm{ÒÏ}}_s{v}_s$

Here, ${\mathrm{ÒÏ}}_s$is density of Styrofoam sphere.

Replace $m_s$with ${\mathrm{ÒÏ}}_s{v}_f$and $v_f$with $\frac{4}{3}{\mathrm{Ï€°ù}}^3$in $mg$$=\left({\mathrm{ÒÏ}}_{\mathrm{f}}{V}_{f}g\right)-m_{s}g$.

$m={\mathrm{ÒÏ}}_f\left(\frac{4}{3}\mathrm{Ï€}{r}^3\right)-{\mathrm{ÒÏ}}_s\left(\frac{4}{3}\mathrm{Ï€}{r}^3\right)$

Substitute $1000kg/m^3$for ${\mathrm{ÒÏ}}_f$,$25cm$for $r$,$150kg/m^3$for ${\mathrm{ÒÏ}}_s$$m=\left(1000\mathrm{kg}/{\mathrm{m}}^3-150\mathrm{kg}/{\mathrm{m}}^3\right)\left[\frac{4}{3}\mathrm{Ï€}(25\mathrm{cm}{)}^3{\left(\frac{1×{10}^{-2}\mathrm{m}}{1\mathrm{cm}}\right)}^3\right]$

$=55.6\mathrm{kg}$

Maximum mass is$56kg$